Numerical modeling of elastomeric seismic isolators for determining force–displacement curve from cyclic loading

modeling of a tank with water and its different applications, its form of modeling is explained, both technical and theoretical, and the analysis of the results of the updated form of investigation of the subject....

ORIGINAL RESEARCH

Numerical modeling of elastomeric seismic isolators for determining

force–displacement curve from cyclic loading

Majid Saedniya 1  · Sayed Behzad Talaeitaba 2

Received: 12 May 2018 / Accepted: 3 August 2019 / Published online: 13 August 2019

© The Author(s) 2019

Abstract

The ideal performance of seismic isolating systems during the past earthquakes has proved them to be very useful in protect‑ ing structures against earthquakes The cyclic loading experimental tests are an important part in the process of completing the design of the isolators, yet they are very expensive and time consuming Using the accurate analytical modeling of hys‑ teresis tests and knowing the limitations and the amount of error of the finite elements model and its effect on designing the isolated structure make it possible to reduce the financial and time expenses involved in designing seismic isolators along with experimental tests In the present study, the cyclic loading of two different isolating systems, namely, the high damping rubber bearing (HDRB) and lead rubber bearing (LRB) have been modeled and analyzed in ABAQUS and the outcomes were compared with the experimental results attained by other researchers Regarding the fact that the most important and compli‑ cated component of the elastomeric isolating system is rubber, it was modeled using various strain energy functions Other factors affecting the finite elements models of elastomeric isolators were also studied After comparing the effective stiffness

of the experimental sample with the analytical model of HDRB, the Yeoh function had the best performance in determining the effective stiffness of the isolating system with an error of less than 7% In studying LRBs, too, three types of bearings with different dimensions and lateral strain values were studied; the polynomial function in shear strain value of 150% had the best performance in estimating effective stiffness and damping with errors of less than 3% and 18%, respectively

Keywords Cyclic loading test · High damping rubber bearing · Lead rubber bearing · Finite element analysis · Strain energy function · Analytical modeling

Introduction

To design an elastomeric isolating system, first the size and

characteristics of the isolator such as its stiffness and effec‑

tive damping are determined based on the type and features

of the structure as well as the instructions in related codes

and the tables suggested by the manufacturers Afterwards,

the first sample of isolators is produced by the manufac‑

turer Regarding the importance of proper performance of

these bearings, they must undergo some cyclic loading tests

so that their force–displacement behavior can be acquired

Among the most important factors that should be reported in the results of such tests are (Naeim and Kelly 1999):

• design displacement;

• effective stiffness in the design displacement;

• amount of energy damping in each cycle at the design displacement

After delivering the accurate values from lab tests, the isolator and the structure’s design are modified These tests, although having lots of significant advantages, are very expensive Moreover, during the communication cycle between the lab and the designers before reaching accept‑ able results, a lot of time and money is spent In the present study, we tried to examine high damping rubber bearings (HDRBs) and lead rubber bearings (LRBs) using the finite element software (ABAQUS) to:

* Sayed Behzad Talaeitaba

talaeetaba@iaukhsh.ac.ir

1 Islamic Azad University of Khomein, Khomein, Iran

2 Islamic Azad University of Khomein, Khomeinishahr Branch

of Azad University, P.O Box: 84175‑119, Khomeyni Shahr,

Iran

1 Investigate the possibility of reducing the expenses of

manufacturing isolators through modeling the hysteresis

cycle tests

2 Exactly know the effective factors in the resulting error

and the contribution of each of them in that

3 Learn about the performance of seismic isolating sys‑

tems before running experimental tests

4 Control the future experimental tests

5 Have the ability to build some new seismic isolators and

model their tests using the results of the present study

In the past, researchers have used numerical methods as

a seismic isolator analysis tool In all of these researches,

the main goal was to obtain precise and inexpensive models

for the analysis of isolators by numerical methods (Asl et al

2014; Ohsaki et al 2015; Mishra et al 2013; Talaeitaba et al

2019)

Finite element modeling

Modeling the seismic isolators using the finite element soft‑

ware program is generally done in two ways In the first,

the whole isolating part and the structures under and over

it are modeled in the form of concentrated mass, spring,

and damper, then the whole system behavior is assessed In

the second method, however, only the isolating system is

modeled and tests on it were performed (Suhara et al 1992;

Martelli et al 1992)

Modeling methods of the parts

At the beginning of the analysis process, each element can

generally be modeled in three forms: two‑dimensional,

three‑dimensional, and axisymmetrical Two‑dimensional

modeling has a lot of limitations and was popular in the past

decades regarding the hardware possibilities of those days

(Imbimbo and De Luca 1998) Most cases of modeling now

are three‑dimensional or axisymmetrical Forni et al state

that although in axisymmetrical models the solution pro‑

cess takes less time, they will not be very accurate for shear

strains of more than 150% and three‑dimensional models

are more efficient for horizontal deformations (Talaeitaba

et al 2019)

Introducing materials

The most important step in modeling an elastomeric isola‑

tor is defining the materials especially rubber In this sec‑

tion, the properties of the materials used in the model are

explained

Steel

The main role of the steel in rubber isolators is preventing high strains under vertical loads The best‑known materials for modeling are metals Steel was defined as an elastoplastic material with characteristics presented in Table 1 (Mori et al

1996; Doudoumis et al 2005)

Lead

Lead has a crystal structure which will change as displace‑ ment increases Lead reaches the yield under shear force

in relatively low tensions of about 8–10 MPa and shows a stable hysteresis behavior and never reaches fatigue due to the repeated yields under the lateral cyclic dynamic loads (Trevor 2001) Lead is defined as an elastoplastic material according to Table 2

Rubber

The elastomeric materials have an almost linear behavior in small strains; however, their behavior is highly non‑linear and elastic in large strains This non‑linear behavior causes the material’s parameters including the shear modulus and elasticity modulus to change as the strain increases (Guo and Sluys 2008) The rubber’s shear modulus and damping depend on the load size, temperature changes, and the strain history (Charlton et al 1993)

In the finite element program, materials whose stress–strain curve in large deformations is non‑linear and elastic are called hyperelastic materials Polymers such as rubber are among these (APASmith 2007)

To model these materials, they can be assumed to be iso‑ tropic, isothermal, elastic, and incompressible The effect of loading frequency and time on their behavior is also ignored (Salomon et al 1999; Venkatesh and Srinivasa Murthy 2012) Hyperelastic materials are described in terms of “strain

energy potential” (U) which defines the strain energy stored

in the material per unit of reference volume (volume in the initial configuration) as a function of the strain at that point

in the material (APASmith 2007)

Table 1 Steel properties (Imbimbo and De Luca 1998)

Table 2 Lead properties (Doudoumis et al 2005)

These functions have the following characteristics (Garcia

et al 2005):

1 The stress–strain function of the model will not change

for frequent loadings

2 The stress–strain function is fully reversible

3 The materials are assumed to be completely elastic with

no permanent deformation

There are several forms of strain energy potentials avail‑

able in Abaqus to model approximately incompressible iso‑

tropic elastomers which are listed below In the presented

equations, I1, I2, and I3 are the deviatoric strain invariants

Strain energy functions are defined by these coefficients

Also, Je1 is the elastic volume ratio (APASmith 2007)

i Mooney–Rivlin form

where U is the strain energy per unit of reference vol‑

ume C01, C10, D1 are the temperature‑dependent material

parameters

ii Neo‑Hookean form

where D1 and C10 are the temperature‑dependent material

parameters

iii Ogden form

where 𝜇 i , 𝛼 i , D i are the temperature‑dependent material

parameters and N is the material parameter.

iv Yeoh form

where C i0 and D i are the temperature‑dependent material

parameters

v Arruda–Boyce form

(1)

U = C10(

I1−3)

+ C01(

I2−3) + 1

D1

(

Jel−1)2

,

(2)

U = C10(

I1−3)

+ 1

D1

(

Jel−1)2

,

(3)

U =

N

∑

i=1

2𝜇 i

𝛼2

i

(𝜆 𝛼 i

1 + 𝜆 𝛼 i

2 + 𝜆 𝛼 i

3 −3) +

N

∑

i=1

1

D i

(

Jel−1)2i

,

(4)

U = C10(

I1−3)

+ C20(

I1−3)2

+ C30(

I1−3)3

+ 1

D1

(

Jel−1)2

+ 1

D2

(

Jel−1)4

+ 1

D3

(

Jel−1)6

,

(5)

U = 𝜇

{

1

2

(

I1−3)

+ 1

20𝜆2

m

(

I12−9) + 11

1050𝜆4

m

(

I31−27)

+ 19

7000𝜆6

m

(

I41−81)

+ 519

613750𝜆8

m

(

I15−243)}

+ 1

D

(

Jel2−1

2 −ln J

el

) ,

where 𝜇 , 𝜆 m and D are the temperature‑dependent material parameters The locking stretch 𝜆 m can be obtained from the

limiting chain stretch ( 𝜆lim) , which is the stretch at which the stress starts to increase without any limit (see Fig. 1

Bergstrom 2002) λ m is determined according to the Eq. (6)

vi Polynomial form

where C ij and D i are the temperature‑dependent material

parameters N is a material parameter.

vii Reduced polynomial form

viii van der Waals form

where

(6)

𝜆 m=

√ 1 3

[

𝜆2lim+ 2

𝜆lim

]

(7)

U =

N

∑

i+j=1

C ij(

I1−3)i(

I2−3)j

+

N

∑

i=1

1

D i

(

Jel−1)2i

,

(8)

U =

N

∑

i=1

C i0(

I1−3)i

+

N

∑

i=1

1

D i

(

Jel−1)2i

(9)

U = 𝜇

{

−(

𝜆2m−3) [ln (1 − 𝜂) + 𝜂] − 2

3a

(I −3 2

)3 2

}

+ 1

D

(

Jel2−1

2 −ln J

el

) ,

(10)

I = ( 1 − 𝛽)I1+ 𝛽I2

(11)

𝜂 =

√

I −3

𝜆2

m−3.

Fig 1 Determining the limiting chain stretch (λlim) (Bergstrom 2002)

In order for the design predictions to be relevant, it is

essential that the materials’ properties are determined under

test conditions appropriate for the service conditions Where

combinations of test data are supplied to derive model coef‑

ficients, these data must be determined at the same tem‑

peratures and strain rates (Garcia et al 2005) These tests

specify the force–displacement relation of the material in

four different modes of deformation which follow (Charlton

et al 1993):

1 Uniaxial tension and compression test

2 Equibiaxial tension and compression test

3 Planar shear test (also known as pure shear)

4 Volumetric tension and compression

All tests must be done on the same material and at the

same temperature The most commonly performed experi‑

ments are uniaxial tension, uniaxial compression, and pla‑

nar tension After running these tests and determining the

strain–stress relation in each of the above modes, the results

are fed into the program and the program matches the results

with the function; then the adapted curve is exhibited and the

required coefficients are determined (APASmith 2007) In

other words, for each of the above tests, the test is simulated

with ABAQUS software, and then the required parameters

of the simulation are extracted

Modeling the high damping rubber bearing

(HDRB)

The high damping rubber bearing under study is selected

from the research done by Yoo et al in the Korea Atomic

Energy Research Institute (Doudoumis et al 2005; Yoo et al

2002)

Geometry

The model is three‑dimensional with the following attributes

presented in Table 3

Defining materials

The high damping rubber bearing consists of two materials: rubber and steel Steel is defined as an elastoplastic mate‑ rial with the properties given in Table 1 To model the rub‑ ber, uniaxial, biaxial, and planar shear tests were carried out (Yoo et al 2002; Busfield and Muhr 2003; Tun Abdul Razak Research Centre 2002; Bradley et al 2001) whose results are presented in Figs. 2 3 and 4 The initial shear modulus for rubber is 0.4 MPa

Regarding rubber tests’ results and by matching curves with each of the strain energy functions, the coefficients for each function were determined as Table 4

Afterwards to specify the amount of error in each model, the value of shear modulus for each function was compared with its experimental value

As seen in Table 5, the best estimation of the initial shear modulus was done by van der Waals and Yeoh functions

After that comes Arruda–Boyce and polynomial (N = 2), Ogden (N = 3), neo‑Hookean and Mooney–Rivlin functions

in order of estimation accuracy Other than the functions of

Table 3 Geometric features of the HDRB (Doudoumis et al 2005)

Fig 2 Rubber’s uniaxial test (Yoo et  al 2002; Busfield and Muhr 2003; Tun Abdul Razak Research Centre 2002; Bradley et al 2001)

Fig 3 Rubber’s biaxial test (Yoo et  al 2002; Busfield and Muhr 2003; Tun Abdul Razak Research Centre 2002; Bradley et al 2001)

Yeoh and polynomial (N = 2) which have estimated the origi‑

nal shear modulus below the real value, all other functions

had a higher estimation than its true measure

Loading

According to the experimental processes on the model, in

the first step, a vertical load of 50 kN was applied to the

model uniformly distributed on top In the second step, as

the load exertion continues, a shear displacement of 60 mm

was applied to the system The amount strain due to this

shear displacement was equal to 200%

Meshing

The Hex element shape was used for meshing the model in ABAQUS C3D8H and C3D8R elements of the software were used to model rubber sheets and steel plates, respec‑ tively The size of the meshes was assigned 4.5 units and the whole enmeshed elements were as many as 18,700 The meshed HDRB and its deformation shape under shear strain are shown in Fig. 5

Solution method

To analyze the finite elements model, the static general anal‑ ysis was used; due to the high amount of displacement, the non‑linear geometry was also activated

The analysis results

After analyzing the models, the numerical hysteresis loops were attained which are shown in Figs. 6 and 7 along with the experimental hysteresis loop for comparison

Fig 4 Rubber’s planar test (Yoo et al 2002; Busfield and Muhr 2003;

Tun Abdul Razak Research Centre 2002; Bradley et al 2001)

Table 4 Coefficients of the

strain energy functions for the

rubber in HDRBs

Table 5 Amount of error in

calculating the model’s initial

shear modulus in comparison to

the experimental value

(MPa) Experimental shear modulus (Salomon et al 1999) (MPa) Error percentage (%)

Fig 5 HDRB model and its deformed shape

As seen in Figs. 6 and 7, in modeling HDRB the resulted

hysteresis loop is linear Therefore, the amount of energy

damping in the isolating system cannot be measured To deter‑

mine the amount of error resulting from each hysteresis loop,

the effective stiffness values of the models were calculated

using Eq. (12) and compared to the experimental results in

Table 6

where F+ is the corresponding force with the maximum

displacement, and F− is the corresponding force with the

minimum displacement

What is concluded from the hysteresis loops and Table 6

is that the effective stiffness of the experimental model is

(12)

Keff= F

+− F−

Δ+− Δ−,

generally less than that of the analytical models The best result is for Yeoh function Arruda–Boyce, van der Waals,

and Ogden (N = 3) functions come next, respectively These

four functions have an error less than 10%

Modeling the lead rubber bearings (LRB)

For lead rubber bearings (LRBs), three samples were mod‑ eled The first model was chosen from the study of Dou‑ doumis et al (2005) The second and third models were selected from the paper presented by Nersessyan et al (2001) The interpretation of the modeling process and the results are shown in the following

Fig 6 Experimental and numer‑

ical hysteresis loops resulted

from Mooney–Rivlin, polyno‑

mial (N = 2), van der Waals and

Yeoh functions for HDRBs

Fig 7 Experimental and numer‑

ical hysteresis loops resulted

from neo‑Hookean, Ogden

(N = 3) and Arruda–Boyce func‑

tions for HDRBs

All three models are three‑dimensional having the charac‑

teristics shown in Table 7

Defining material

LRBs consist of three main materials: rubber, steel, and lead

Steel was defined according to Table 1 and lead according to

Table 2 The rubber’s properties are listed in Table 8

Regarding the fact that there are not any experimental

results for quadruple tests on rubber for determining the

coefficients of the strain energy functions, the required tests

were done by the finite element software itself To do this,

the software instructions state that there must be the results

of at least two tests Regarding the fact that rubber is usu‑

ally considered incompressible, there is no need to do the

volumetric test In this study, the tests that have been done

for rubber are the uniaxial and planar shear tests

To do the uniaxial tension test, the rubber whose proper‑ ties are presented in the standard DIN53504‑S2 was used The dimensions of the model are shown in Fig. 8 (Trevor

2001) The model’s thickness is 2 mm

The sample was modeled two dimensionally and the rubber was defined using the Arruda–Boyce function To use this function, the initial shear modulus and the amount

of locking stretch ( 𝜆 m ) are needed Regarding the amount

of isolator’s strain and the vast range of numerical tests,

3 seems the proper value for 𝜆 m This amount, which was determined by trial and error, was a premise that the speed and precision of the solution would be higher

To analyze the model, the implicit dynamic analysis was used, and the model was considered as planar tension Therefore, the CPS4R element was used for meshing In Fig. 9, the created model of rubber as well as its deformed shape under uniaxial tension is presented

After analyzing the model, the strain–stress curve for rub‑ ber in its central zone was attained

Table 6 Comparing the

calculated effective stiffness

with the experimental results in

HRDBs

Table 7 Geometrical specifications of the LRB models

Luca 1998) Second model (Bergstrom 2002) Third model (Bergstrom

2002)

To do the planar shear test, a two‑dimensional model for

the rubber with the following measurements and a thickness

of 2 mm was used (Forni et al 2002) (Fig. 10)

All the stages of modeling and analysis of this model are

the same as those of the rubber under uniaxial tension test The

meshed model and its deformed shape are shown in Fig. 11

After attaining the stress–strain curves according to

Fig. 12 the coefficients for strain energy functions were

determined according to Table 9 for the first type of rubber

in LRBs and Table 10 for second and third types

After determining the coefficients of the strain energy

functions, the initial shear modulus value was compared to

the experimental value in order to calculate the amount of

error in each function (Table 11)

For the first rubber model, the Arruda–Boyce function

had the least error followed by the functions of Yeoh, Ogden

(N = 3), neo‑Hookean, Mooney–Rivlin, van der Waals and

polynomial (N = 2), respectively Other than the functions of

Mooney–Rivlin and neo‑Hookean, all other functions esti‑

mated the initial shear modulus below the real value

For the second and third models of rubber, too,

the Arruda–Boyce function had the least error After

that came the functions of Yeoh, van der Waals,

Ogden (N = 3), polynomial (N = 2), neo‑Hookean and

Mooney–Rivlin The functions of Mooney–Rivlin, neo‑

Hookean and Yeoh have estimated the initial value of the

shear modulus higher than its real value, yet the other functions have reached a lower value than the real one

Loading

To conduct the hysteresis cycle test, first the vertical load

is applied to the model Then, in the second stage, along with this load, the shear displacement is exerted on the isolating system The amount of the vertical load and dis‑ placement applied to each of the three models of LRB is shown in Table 12

Meshing

The Hex element shape was used for meshing the model in ABAQUS C3D8H and C3D8R elements of the software were used to model rubber sheets and steel plates, respec‑ tively All the meshed elements of the first model were equal to 6016 elements, the second model 30349, and the third model 4945 The meshed model of each LRB and its deformed shape caused by the shear displacement are pre‑ sented in Figs. 13, 14 and 15

Table 8 Initial properties of

rubber in the modeled LRBs Rubber properties First model (Dou‑doumis et al 2005) Second model (Nersessyan et al 2001) Third model (Nersessyan et al

2001)

Fig 8 Dimensions of the rubber model for uniaxial tension test (mm)

Fig 9 Rubber model for uniaxial tension test and its deformed shape under tension force

Fig 10 Dimensions of the rubber model for the planar shear test (mm)

Analysis results

In Figs. 16, 17, 18, 19, 20 and 21, the hysteresis loops of the

strain energy functions are compared with the experimental results

First model:

Fig 11 Rubber model for the planar shear test and its deformed shape under tension force

Fig 12 Stress–strain curves resulted from rubber’s tests

The second model:

The third model:

In these hysteresis loops, the models’ behaviors have been

illustrated qualitatively After calculating the effective stiffness

and the effective damping for each hysteresis loop, their quantitative errors are determined The value of the effective stiffness and the effective damping can be determined using

Table 9 Coefficients of the

strain energy functions for the

first type of rubber in LRBs

Table 10 Coefficients of the

strain energy functions for the

second and third types of rubber

in LRBs

Table 11 Amount of error in calculating the initial shear modulus of the numerical models using the functions in comparison to the experimen‑ tal value in LRBs

lus (MPa) Experimental shear modulus (MPa) (Martelli et al 1992; Garcia et al 2005) Error percentage

(%)